STUDY OF IMAGE DENOISING USING CURVELET TRANSFORM …ethesis.nitrkl.ac.in/5164/1/109CS0633.pdf · STUDY OF IMAGE DENOISING USING CURVELET TRANSFORM RAJAT SINGH ... = s(x, y) + n(x,

STUDY OF IMAGE DENOISING USING

CURVELET TRANSFORM

RAJAT SINGH (109CS0345)

D.S. MEENA (109CS0633)

Department of Computer Science and Engineering

National Institute of Technology, Rourkela

STUDY OF IMAGE DENOISING USING CURVELET

TRANSFORM

A Thesis submitted in partial fulfillment of the requirements for the degree

Of

Bachelor of Technology in “Computer Science and Engineering”

By


D.S. MEENA (109CS0633)

Under guidance of :

Prof. Ratnakar Dash

Prof. R.K. Mohapatra

Department of Computer Science and Engineering

National Institute of Technology

Rourkela-769008 (ODISHA)

2013

- 1 -

DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING

NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA

ODISHA, INDIA-769008

CERTIFICATE

This is to certify that the thesis entitled “Image de-noising using curvelet transform”, submitted

by Rajat Singh (Roll No. 109CS0345) and Devendra Singh Meena (Roll No. 109CS0633) in partial

fulfillment of the requirements for the award of Bachelor of Technology in Computer Science and

Engineering during session 2012-2013 at National Institute of Technology, Rourkela. A bonafide

record of research work carried out by them under my supervision and guidance.

The candidates have fulfilled all the prescribed requirements.

The thesis which is based on candidates’ own work, have not submitted elsewhere for a

degree/diploma.

In my opinion, the thesis is of standard required for the award of a bachelor of technology degree in

Computer Science and Engineering.

Place: Rourkela

Dept. of Computer Science Engineering Prof. Ratnakar Dash National institute of Technology Prof. R.K. Mohapatra Rourkela-769008

- 2 -

ACKNOWLEDGEMENT

We are grateful to The Department of Computer Science and Engineering for giving us the

opportunity to carry out this project, which is an integral fragment of the curriculum in B. Tech

programme at the National Institute of Technology, Rourkela. We would like to express our heartfelt

gratitude and regards to our project guide, Prof. Ratnakar Dash and Prof. R.K. Mohapatra,

Department of Computer Science and Engineering, for being the corner stone of our project. It was

their incessant motivation and guidance during periods of doubts and uncertainties that has helped us

to carry on with this project. We would like to thank Prof. A.K. Turuk, Head of the Department, and

Computer Science and Engineering for his guidance, support and direction. We are also obliged to

the staff of Computer Science and Engineering Department for aiding us during the course of our

project. We offer our heartiest thanks to our friends for their help in collection of data samples

whenever necessary. Last but not the least, we want to acknowledge the contributions of our parents

and family members, for their constant and never ending motivation.


DEVENDRA SINGH MEENA (109CS0633)

B.TECH (COMPUTER SCIENCE AND ENGINEERING)

a

Dedicated to

Our beloved parents

b

ABSTRACT

The images usually bring different kinds of noise in the process of receiving, coding and

Transmission. In our implementation the Curvelet transform is used for de-noising of image.

Two digital implementations of the Curvelet transform the Unequally Spaced Fast Fourier

Transform (USFFT) and the Wrapping Algorithm are used to de-noise images degraded by

different types of noises such as Gaussian, Salt and Pepper, Random, Speckle and Poisson

noise. This thesis aims at the effect the Curvelet transform has in Curve-let shrinkage assuming

different types of noise models. A signal to noise ratios a measure of the quality of de-noising

was preferred. The experimental results show that the normal Curvelet shrinkage approach fails

to remove Poisson noise in medical images.

Keywords Curvelet transform, wrapping algorithm, USFFT

i

CONTENTS Abstract i

Contents ii

CHAPTER 1

INTRODUCTION

1.1 Introduction 2

CHAPTER 2

ADDITIVE AND MULTIPLICATIVE NOISES

2.1 Guassian noise 5

2.2 Salt and pepper noise 6

2.3 Speckle noise 7

2.4 Brownian noise

2.5 Poisson noise

CHAPTER-3

THE WAVELET TRANSFORM AND DENOISING

3.1 Discrete wavelet transform – Principles 10

3.2 Properties of DWT 12

3.3 Wavelet thresholding 13

CHAPTER-4

THE CURVELET TRANSFORM

4.1 Introduction 16

4.2 Image de-noise using curve-lets 17

4.3 Algorithm 17

4.4 Curve-let shrinkage/thresholding 18

CHAPTER-5

RESULTS AND DISCUSSIONS

5.1 RESULTS 19 ii

CHAPTER-6

CONCLUSION

6.1 Conclusion 24

References 26

iii

CHAPTER1

Introduction

1

Introduction

A very vast portion of digital image processing is concerned with image de-noising. This includes

research in algorithm and routine goal oriented image Processing. Image restoration is the removal

or reduction of degraded images that are Incurred while the image is being obtained. Degradation

comes from blurring as Well as noise due to various sources. Blurring is a form of bandwidth

reduction in the image caused by the imperfect image formation process like relative motion between

the camera & the object or by an optical system which is out of the focus. When aerial photographs

are taken for remote sensing purposes, atmospheric turbulence introduces blurs, optical system

aberration and relative motion between camera and the ground. With these blurring effects, the

recorded image can also be corrupted by noises. A noise can be introduced in the transmission

medium due to a noisy channel, errors during the measurement process and during quantization of

the data for digital storage. Each element in the imaging chain such as film, lenses, digitizer, etc.

contribute to the degradation. Image de-noising is often used in the field of photography or publishing

where an image is somehow degraded but it needs to be improved before it can be printed. For this

type of application we need to know about the degradation process in order to design a model for it.

When a model for the degradation process is designed, the inverse process can be applied to the image

to de-noise it back to its original form. It is the type of image de-noise often used in space exploration

to help eliminate artifacts generated by mechanical jitter in a spacecraft or to reduce distortion in the

optical system of a telescope. Image de-noising finds applications in fields such as astronomy where

the resolution limitations are high, in medical imaging where the physical requirements for high

quality imaging are needed for analyzing the images of unique events and in the forensic science

where potentially useful photographic information is sometimes of extremely bad quality.

2

Digital images are 2-D matrices in image processing and important task is to adjust values of these

matrices in order to get clear features of images. The adjusting of values obeys a certain mathematical

model. The main challenge is to build suitable mathematical models for practical requirements.

Taking image de-noising example many mathematical models are based on a frequency partition of

the image, where components having high frequency are interpreted as noise which have to be

removed while those with low frequency can be seen as features to be remained. Curve-lets, which

we are using, can be seen as an effective model that not only considers a multi scale time-frequency

local-partition but also makes use of the direction of features.

Applications of wavelets are increasingly being used in scientific and engineering fields, traditional

wavelets do well only at representing point singularities, as they ignore the geometric properties of

structures and do not exploit the regularity of edges. Thus, de-noising, wavelet based compression

or structure extraction become computationally inefficient for geometric features with line and

surface singularities. For ex, when we download compressed image or video, we mostly find a mosaic

phenomenon. The mosaic phenomenon comes from the poor ability of wavelets to handle line

singularities. In fluid mechanics, discrete wavelet thresholding mostly leads to oscillations along

edges of the coherent eddies, and to the deterioration of the vortex tube structures, which later can

cause an unphysical leak of energy into neighboring scales producing an artificial “cascade" of

energy.

3

CHAPTER2

Additive and Multiplicative

Noises

4

Noise is undesired information that degrades the image. In the image de-noising process, information

of the type of noise present in the original image plays a significant role. Mostly images can be

corrupted with noise modeled with either a uniform, Gaussian, or salt and pepper distribution.

Another type of noise is a speckle noise which is multiplicative in nature. Noise is present in image

either in an additive or multiplicative form.

Rule for additive noise

w(x, y) = s(x, y) + n(x, y) , ……………(1)

Rule for multiplicative noise

w(x, y) = s(x, y)×n(x, y) , ……………(2)

where (x,y) is original signal, n(x,y) is the noise introduced into the signal to produce a noisy image

w(x,y), and (x,y) is the pixel location. The above image algebra is done at pixel level. Image addition

also has applications in image morphing. Image multiplication means the brightness of the image is

varied.

The digital image acquisition process transforms an optical image into a continuous electrical signal

that is, sampled. In every step of the process there are fluctuations caused by natural phenomena,

adding random value to the exact brightness value for a given pixel.

2.1 Gaussian Noise

A gaussian noise is evenly distributed in the signal. That means every pixel in the noisy image is the

sum of the a random Gaussian distributed noise value and true pixel value. This type of noise has a

Gaussian distribution, which has a probability distribution function given by,

5

……….(3)

Where g represents the gray level, m is the mean of the function and σ is standard deviation in the

noise. Graphically, it is represented as shown in Figure below.

Figure 1 : Gaussian distribution

2.2 Salt and Pepper Noise

Salt and pepper noise is impulse type of noise, that is also referred to as intensity spikes. It is caused

generally due to the errors in the data transmission. It has only two possible values that is a and b.

The probability of each is typically less than 0.1. Corrupted pixels can be set alternatively to the

minimum or to the maximum value, giving image a “salt and pepper” like appearance. Pixels remain

unchanged for unaffected. For a 8-bit image, the value of pepper noise is 0 and for salt noise 255.

Salt and pepper noise is mainly caused by malfunctioning of pixel elements in the sensors of cameras,

faulty memory locations, or timing errors of the digitization process.

6

The probability density function for this type of noise is shown in the Figure below.

. Figure 2: PDF for salt and pepper noise

2.3 Speckle Noise

Speckle noise is multiplicative noise. This type of noise occurs mostly in all coherent imaging

systems such as acoustics, laser, acoustics and SAR (Synthetic Aperture Radar) imagery. Source of

this noise is attributed to the random interference between the coherent returns. A fully developed

speckle noise has the characteristic of multiplicative noise. Speckle noise follows a gamma

distribution and given as

………………(4)

where variance is a2αand g is the gray level.

7

2.4 Brownian Noise

Brownian noise is under the category of fractal or 1/f noises. The mathematical model for 1/f noise is

the fractional Brownian motion. Brownian motion is non-stationary stochastic process which follows

a normal distribution. Brownian noise is a special case of 1/f noise. It can be obtained by integrating

white noise. It can be graphically represented as shown in Figure below.

Figure 3 : Brownian noise distribution

2.5 Poisson Noise

Many images like as those from radiography, have noise that satisfies a poisson distribution.

Magnitude of Poisson noise varies across an image and it depends on the image intensity, thatmakes

removing such noise very hard. Poisson images occur in situations where the image acquisition is

performed using the detection of particles (e.g) counting photons being emitted from a radioactive

source is applied in medical imaging like SPECT and PET , therefore Poisson noise reduction is an

essential problem.Poisson noise is generated from the data in place of adding artificial noise to the

data. For ex. if a pixel in an unsigned integer input has the value 10, then corresponding output pixel

will be generated from Poisson distribution with a mean 10.

8

CHAPTER3

The Wavelet Transforms and De-noising

9

3.1. Discrete Wavelet Transform (DWT) - Principles:

Wavelets are the mathematical functions which analyze data according to the scale or

resolution. They help in studying a signal in different windows or in different resolutions. For

example, if the signal is viewed in the large window, gross feature can be noticed, and if viewed

in a small window, only the small features can be noticed. The wavelets provide some

advantages over Fourier transforms. For instance, they do a great job in approximating signals

with sharp spikes and signals having discontinuities. Wavelets can also model music, speech,

video and non-stationary stochastic signals. The wavelets can be used in applications such as

turbulence, image compression, human vision, earthquake prediction, etc.

The term “wavelets” is refered to a set of orthonormal basis functions generated by translation

and dilation of scaling function φand a mother wavelet ψ. A finite scale multi resolution

representation of a discrete function is called as a discrete wavelet transform. DWT is a fast

linear operation on the data vector, whose length is an integeral power of 2. This transform is

orthogonal and invertible where the inverse transform expressed as the matrix is the transpose

of the transform matrix. The wavelet basis or function, unlike sines and cosines in Fourier

transform, is localized in space. Similar to sines and cosines the individual wavelet functions

are localized in frequency.

The orthonormal basis or wavelet basis is defined as

……………(5)

The scaling function is given as

…………. (6)

10

Where ψ is the wavelet function and j and k are integers that scale and dilate the wavelet

function. Factor ‘j’ in Equations is called as the scale index, which indicates the width of the

wavelet. The location index k provides the position. The wavelet function is dilated by powers

of two and is translated by the integer k. In terms of the wavelet coefficients, the wavelet

equation is

………….(7)

Where g0, g1, g2…. are high pass wavelet coefficients. The scaling equation in terms of scaling

coefficients as given below

………….(8)

The function φ(x) is the scaling function and the coefficients h0, h1,….. are low pass scaling

coefficients. The wavelet and scaling coefficients are related by the quadrature mirror

relationship, which is

………….(9)

Where N is the number of vanishing moments.

11

3.2 Properties of DWT

Properties of a discrete wavelet transforms are given below.

DWT is a fast linear operation, that can be applied on data vectors having length as

integeral power of 2.

DWT is invertible and orthogonal. The scaling function φand the wavelet function ψ

are orthogonal to each other in L2(0, 1), i.e., <φ, ψ>= 0.

The wavelet basis is localized in the space and frequency.

The coefficients satisfies some constraints

………………………(10) & (11)

Here δis the delta function and l is the location index.

…………….(12)

12

3.3 Wavelet Thresholding

The term wavelet thresholding is defined as decomposition of the data of image into wavelet

coefficients, comparing the detailed coefficients having a given threshold value, and

minimizing these coefficients close to zero to remove the effect of noise in the data. Then image

is reconstructed from modified coefficients. This is also known as inverse discrete wavelet

transform. At the time of thresholding, a wavelet coefficient is compared to the given threshold

and is set to zero if its magnitude is less than the threshold otherwise, it is then retained or

modified depending on the thresholding rule. Thresholding distinguishes between coefficients

due to noise and the ones consisting of important signal information. The selection of a

threshold is an important point of interest. It plays an important role in the removal of noise in

the images because de-noising most frequently produces smoothed images, by reducing the

sharpness of the image. Care should be taken to preserve the edges of the de-noised image.

Various methods for wavelet thresholding exists, which rely on the choice of a threshold value.

Typically used methods for image noise removal include Sureshrink, VisuShrink and

BayesShrink. It is necessary to know about the two generic categories of thresholding. These

are hard thresholding and soft thresholding. The hard-thresholding TH is given as

……………(13)

where t is the threshold value. A plot of TH is shown in Figure below

Figure 4 : Hard thresholding

13

Therefore, all coefficients whose magnitude is greater than the selected threshold value T

remains same and the others with magnitudes smaller than t are set to zero. It creates a region

around 0 where the coefficients are considered to be negligible.

Soft thresholding is that where the coefficients with greater than the threshold are shrunk

towards zero after comparing them to the threshold value. It is defined as below

……………..(14)

Figure 5 : Soft Thresholding

Practically, it can be seen that the soft method is much better and yields more visually pleasant

images. This is because the hard method is discontinuous and yields abrupt artifacts in the

images recovered. Also, the soft method yields a smaller MSE (minimum mean squared error)

compared to hard form of thresholding.

14

CHAPTER4

The Curvelet Transform

15

4.1 Introduction

The Curvelet transform is a higher dimensional generalization of the wavelet transform designed to

represent images at different scales and different angles. Curvelet transform is a special member of

the multi scale geometric transforms. It is a transform with a multi scale pyramid with many directions

at each length scale. Curvelets will be superior over wavelets in following cases:

Optimal sparse representation in object with edges

Optimal image reconstruction of severely ill-posed problems

Optimal sparse representation in wave propagators

The idea of the Curve-let transform is first to decompose the image into sub-bands, i.e. to separate

the object into a series of disjoint scales. Curve-lets are initially introduced by Candes and Donoho.

The Discrete Curvelet transform (DCT) takes as input a Cartesian grid of the form ,

, 0 , and outputs a collection of coefficients cD ( j, l, k)

defined by

…………….(15)

where are digital curvelet waveforms which preserve the listed properties of the continuous

curve-let. DCT can be implemented in two ways. First method is based on unequally-spaced fast

Fourier transform (USFFT) and the second is based on the Wrapping of specially selected Fourier

samples. The two implementations are differ by spatial grid used to translate curve-lets at each scale

and angle.

16

The curve-let de-noising method consists of the following steps:

Estimate the noise standard deviation ‘σ’ in the input image

Calculate the Curve-let transform of the input image. We get a set of bands wj, each band wj

contains Nj coefficients and corresponds to a given resolution level.

Calculate noise standard deviation’ ’ for each band j of the Curve-let transform.

For each band j do : Calculate the maximum of the band and multiply each curve-let

coefficient.

Reconstruct the image from the modified curvelet Coefficient

4.2 Image de-noising using curve-lets

Image De-noising is implemented to produce good estimates of the original image from noisy

observations. The restored images hould contain less noise than the observations while still keep

sharp transitions (i.e edges). Suppose an image f(m,n) is corrupted by the additive noise

g(m,n) = f(m,n) + η(m,n) …………….(16)

where η(m,n) are independent identically distributed Gaussian random variable with zero mean and

variance σ2. Image de-noising algorithms vary from simple thresholding to complicate model based

methods. However simple thresholding methods can remove most of the noise.

4.3 Algorithm

1. Apply the Forward Curvelet transform to the noisy image.

2. Threshold the Curvelet co-efficients to remove some insignificant curve-let co-efficients by using

a thresholding function in the curve-let domain.

3. Inverse Curvelet transform of the thresholded co-efficients to reconstruct a function.

17

4.4 Curve-let Shrinkage/Thresholding

Shrinkage/thresholding plays an important role incurve-let application. Various thresholding

techniques have been applied on the curve-let co-efficient of the observed image. The small co-

efficients are dominated by noise, while co-efficient with large absolute value carry more signal

information than noise. As a result noisy co-efficients (small co-efficients below acertain threshold

value) are replaced by zero.

The curve-let shrinkage is taken as

…………(17)

T denotes the Curve-let transform, T-1 the inverse transform and Sσ is the thresholding function.

18

CHAPTER5

Results and Discussion

19

5.1 Results

The experiments are performed on several types of gray scale images of size 256 x 256 in MATLAB

platform and the Curve-let transform via USFFT and Wrapping technique was Implemented based

on curve-let software package. The effectiveness of the Curve-let shrinkage methods as well as partial

reconstruction of Curve-let coefficients are demonstrated for various gray scale images with a noise

factor of σ = 20 for Random noise, noise density of 0.04 for Salt & Pepper noise, a multiplicative

noise factor of 0.05 for Speckle noise and for Gaussian white noise a mean of 0 and 0.01 variance is

used.

To compare the results of different curvelet thresholding techniques, PSNR measure is used

…………..(18)

here fmax is the maximum value of the image intensities.

The average PSNR measurements of various gray scale images degraded by Random noise, Speckle

noise, Salt & Pepper noise, Gaussian noise and Poisson noise with the two digital implementations

of Curvelet transform are listed in tables.

Table1: De-noising results (PSNR in dB) with curve-let via

USFFT for the four de-noising methods for CT images

Noise Noisy image Soft

Thresholding

Hard

Thresholding

Partial

reconstruction

Random 21.09 26.89 28.90 24.28

Salt and pepper 18.67 24.70 20.87 19.82

Guassian 19.77 27.70 28.12 22.40

Speckle 21.20 27.12 28.90 22.50

Poisson 27.55 27.28 31.24 28.10

20


USFFT for the four de-noising methods for MRI images


Thresholding

Hard

Thresholding

Partial

reconstruction

Random 22.29 26.67 28.30 24.50

Salt and pepper 16.47 22.74 19.87 17.34

Guassian 20.77 25.76 25.12 22.23

Speckle 27.20 26.42 28.39 27.28

Poisson 33.56 27.55 28.54 29.10


USFFT for the four de-noising methods for satellite images


Thresholding

Hard

Thresholding

Partial

reconstruction

Random 22.70 29.67 33.50 23.30

Salt and pepper 18.56 23.82 19.05 18.10

Guassian 21.80 26.36 26.62 23.23

Speckle 25.40 30.83 33.45 27.76

Poisson 32.90 30.55 33.67 32.34


Wrapping for the four de-noising methods for CT images


Thresholding

Hard

Thresholding

Partial

reconstruction

Random 22.05 27.56 29.54 22.30

Salt and pepper 17.36 22.80 19.05 17.26

Guassian 21.96 24.46 27.12 21.24

Speckle 22.51 24.53 27.67 22.53

Poisson 29.85 25.68 30.10 30.05


Wrapping for the four de-noising methods for MRI images


Thresholding

Hard

Thresholding

Partial

reconstruction

Random 22.66 26.78 28.54 22.30

Salt and pepper 16.46 21.64 18.67 16.26

Guassian 21.63 24.56 27.33 23.54

Speckle 28.24 28.98 28.69 28.33

Poisson 34.50 26.23 30.16 34.25

21


Wrapping for the four de-noising methods for satellite images


Thresholding

Hard

Thresholding

Partial

reconstruction

Random 22.23 30.48 28.55 22.36

Salt and pepper 16.24 23.40 19.33 16.25

Guassian 21.47 26.55 26.35 22.19

Speckle 25.45 30.90 28.70 28.99

Poisson 32.66 30.60 28.15 34.40

De-noising results for CT image (Random Noise) via

USFFT (a) Original (b) Noisy (PSNR=22.05) (c) Soft

(PSNR=26.48) (d) Hard (PSNR=29.94) (e) Garrote

(PSNR=29.79) (f) Partial Reconstruction (PSNR=24.42)

22

De-noising results for Satellite image (Speckle Noise)

via USFFT (a) Original (b) Noisy (PSNR=26.82) (c)

Soft (PSNR=33.77) (d) Hard (PSNR=36.67) (e)

Garrote (PSNR=36.48) (f) Partial Reconstruction

(PSNR=27.75)

De-noising results for MRI image (Gaussian) via

Wrapping (a) Original (b) Noisy (PSNR=21.69) (c) Soft

(PSNR=23.32) (d) Hard (PSNR=25.27) (e) Garrote

(PSNR=24.79) (f) Partial Reconstruction (PSNR=22.17)

23

CHAPTER7

Conclusion

24

7.1 Conclusions and Discussion:

A strategy for digitally implementing the Curve-let transform via USSFT and Wrapping

technique is presented. The Curve-let transform is capable of resolving 2D singularities and

represents edges more efficiently in images. The resulting implementations have the exact

reconstruction property and gives table reconstruction. The experimental results showed that

the Curve-let transform implemented with USFFT & Wrapping Algorithm using Hard

&Garrote thresholding consistently produces the highest PSNR for Bio medical images,

satellite images & natural images degraded by Random noise, Gaussian noise and Speckle

noise, while soft thresholding offers best results for de-noising Salt & Pepper noise for all the

test images used.

Alhough best results are obtained while de-noising the test images corrupted by Random noise,

Speckle noise, Gaussian noise and Salt & Pepper noise in terms of PSNR gain it is noticed in the

result that lowest PSNR gain is obtained for Bio-medical images when compared to satellite

images.

The results clearly indicates that Curve-let transform using thresholding techniques proves to be

inferior in de-noising Bio-medical images corrupted by Random noise, Gaussian noise, Speckle

noise and Salt and Pepper noise. Curve-let thresholding techniques proves to be a failure in the

removal of Poisson noise in Bio-medical images. Partial reconstruction of curve-let coefficients

proves to be a failure for all types of noises tested with various images. However the visual quality

of images is preserved.

25

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STUDY OF IMAGE DENOISING USING CURVELET TRANSFORM …ethesis.nitrkl.ac.in/5164/1/109CS0633.pdf · STUDY OF IMAGE DENOISING USING CURVELET TRANSFORM RAJAT SINGH ... = s(x, y) + n(x,

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